One of the angles of a right-angled triangle is `15^(@)`, and the hypotenuse is 1 metre. The area of the triangle (in square cm). Is

Options

A. 1220
B. 1200
C. 1250
D. 1215

Answer

To solve the problem step by step, we will calculate the area of the right-angled triangle with one angle measuring \(15^\circ\) and a hypotenuse of 1 meter.

### Step 1: Convert the hypotenuse from meters to centimeters
Since the area needs to be in square centimeters, we first convert the hypotenuse from meters to centimeters.

\[ 1 \text{ meter} = 100 \text{ centimeters} \]

### Step 2: Identify the sides of the triangle
In a right-angled triangle, we can use trigonometric ratios to find the lengths of the other two sides (the opposite and adjacent sides) using the hypotenuse.

Let:
- \( AC \) be the side opposite the \(15^\circ\) angle.
- \( AB \) be the side adjacent to the \(15^\circ\) angle.
- \( BC \) be the hypotenuse.

Using the sine and cosine functions:
\[ AC = BC \cdot \sin(15^\circ) \]
\[ AB = BC \cdot \cos(15^\circ) \]

Substituting \( BC = 100 \) cm:
\[ AC = 100 \cdot \sin(15^\circ) \]
\[ AB = 100 \cdot \cos(15^\circ) \]

### Step 3: Calculate the lengths of \( AC \) and \( AB \)
Using the values of sine and cosine for \(15^\circ\):
- \( \sin(15^\circ) \approx 0.2588 \)
- \( \cos(15^\circ) \approx 0.9659 \)

Calculating:
\[ AC = 100 \cdot 0.2588 \approx 25.88 \text{ cm} \]
\[ AB = 100 \cdot 0.9659 \approx 96.59 \text{ cm} \]

### Step 4: Calculate the area of the triangle
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take \( AB \) as the base and \( AC \) as the height.

Substituting the values:
\[ A = \frac{1}{2} \times AB \times AC = \frac{1}{2} \times 96.59 \times 25.88 \]

Calculating the area:
\[ A \approx \frac{1}{2} \times 96.59 \times 25.88 \approx 1249.56 \text{ cm}^2 \]

### Step 5: Final Result
Thus, the area of the triangle is approximately:
\[ \text{Area} \approx 1249.56 \text{ cm}^2 \]